THE COLORED JONES POLYNOMIALS AND THE ALEXANDER POLYNOMIAL OF THE FIGURE-EIGHT KNOT

Hitoshi Murakami (Japan)

Received February 28, 2007

Abstract

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight knot the series converges in some cases and the limit equals the inverse of its Alexander polynomial.

Keywords and phrases: figure-eight knot, colored Jones polynomial, Alexander polynomial, volume conjecture.